Hypersphere and the Third Laplace-Beltrami Operator

Yıl 2022, , 727 - 732, 30.06.2022

Erhan Güler

https://doi.org/10.17798/bitlisfen.1109645

Öz

In this work, we examine the differential geometric objects of the hypersphere h in four dimensional Euclidean geometry E^4. Giving some notions of four dimension, we consider the ith curvature formulas of the hypersurfaces of E^4. In addition, we reveal the hypersphere satisfying ∆^III h=Ah for some 4×4 matrix A.

Anahtar Kelimeler

Euclidean space, Third Laplace-Beltrami operator, Hypersphere, Gauss map, Curvature

Kaynakça

• [1] M. Obata, “Certain conditions for a Riemannian manifold to be isometric with a sphere,” J. Math. Soc. Japan, vol. 14, pp. 333-340, 1962.
• [2] T. Takahashi, “Minimal immersions of Riemannian manifolds,” J. Math. Soc. Japan, vol. 18, pp. 380-385, 1966.
• [3] S. S. Chern, M. P. do Carmo, and S. Kobayashi, Minimal Submanifolds of a Sphere with Second Fundamental Form of Constant Length, Functional Analysis and Related Fields. Springer, Berlin, 1970.
• [4] S. Y. Cheng and S. T. Yau, “Hypersurfaces with constant scalar curvature,” Math. Ann., vol. 225, pp. 195-204, 1977.
• [5] B. Y. Chen, “On submanifolds of finite type,” Soochow J. Math., vol. 9, pp. 65-81, 1983.
• [6] B. Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, World Scientific, Singapore, 1984.
• [7] B. Y. Chen, Finite Type Submanifolds and Generalizations, University of Rome, 1985.
• [8] B. Y. Chen, “Finite type submanifolds in pseudo-Euclidean spaces and applications,” Kodai Math. J., vol. 8, no. 3, pp. 358-374, 1985.
• [9] B. Y. Chen, “Surfaces of finite type in Euclidean 3-space,” Bull. Math. Soc. Belg., vol. 39, pp. 243-254, 1987.
• [10] B. Y. Chen, “Null 2-type surfaces in E^3,” Kodai Math. J., vol. 11, pp. 295-299, 1988.
• [11] B. Y. Chen and P. Piccinni, “Submanifolds with finite type Gauss map,” Bull. Austral. Math. Soc., vol. 35, pp. 161-186, 1987.
• [12] O. J. Garay,, “An extension of Takahashi's theorem,” Geom. Dedicata, vol. 34, pp. 105-112, 1990.
• [13] U. Dursun, “Hypersurfaces with pointwise 1-type Gauss map,” Taiwanese J. Math., vol. 11, no. 5, pp. 1407-1416, 2007.
• [14] A. Ferrandez,, O. J. Garay and P. Lucas, On a certain class of conformally at Euclidean hypersurfaces, In Global Analysis and Global Differential Geometry, Springer, Berlin, 48-54. 1990.
• [15] M. Choi and Y. H. Kim, Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map, Bull. Korean Math. Soc., vol. 38, pp. 753-761, 2001.
• [16] O. J. Garay, “On a certain class of finite type surfaces of revolution,” Kodai Math. J., vol. 11, pp. 25-31, 1988.
• [17] F. Dillen, J. Pas and L. Verstraelen, “On surfaces of finite type in Euclidean 3-space,” Kodai Math. J., vol. 13, pp. 10-21, 1990.
• [18] S. Stamatakis and H. Zoubi, “Surfaces of revolution satisfying ∆^III x=Ax,” J. Geom. Graph., vol. 14, no. 2, pp. 181-186, 2010.
• [19] B. Senoussi and M. Bekkar, “Helicoidal surfaces with ∆^J r=Ar in 3-dimensional Euclidean space,” Stud. Univ. Babeş -Bolyai Math., vol. 60, no. 3, pp. 437-448, 2015.
• [20] D. S. Kim, J. R. Kim and Y. H. Kim, “Cheng-Yau operator and Gauss map of surfaces of revolution,” Bull. Malays. Math. Sci. Soc., vol. 39, no. 4, pp. 1319-1327, 2016.
• [21] C. Moore, “Surfaces of rotation in a space of four dimensions,” Ann. Math., vol. 21, pp. 81-93, 1919.
• [22] C. Moore, “Rotation surfaces of constant curvature in space of four dimensions,” Bull. Amer. Math. Soc., vol. 26, pp. 454-460, 1920.
• [23] Th. Hasanis, and Th. Vlachos, “Hypersurfaces in E^4 with harmonic mean curvature vector field,” Math. Nachr., vol. 172, pp. 145-169, 1995.
• [24] Q. M. Cheng and Q. R. Wan, “Complete hypersurfaces of R^4 with constant mean curvature,” Monatsh. Math., vol. 118, pp. 171-204, 1994.
• [25] Y. H. Kim and N. C. Turgay, “Surfaces in E^4 with L_1-pointwise 1-type Gauss map,” Bull. Korean Math. Soc., vol. 50, no. 3, pp. 935-949, 2013.
• [26] K. Arslan, B. K. Bayram, B. Bulca, Y.H. Kim, C. Murathan and G. Öztürk, “Vranceanu surface in E^4 with pointwise 1-type Gauss map,” Indian J. Pure Appl. Math., vol. 42, no. 1, pp. 41-51, 2011.
• [27] K. Arslan, B. K. Bayram, B. Bulca and G. Öztürk, “Generalized rotation surfaces in E^4,” Results Math., vol. 61, no. 3, pp. 315-327, 2012.
• [28] E. Güler, M. Magid and Y. Yaylı, “Laplace-Beltrami operator of a helicoidal hypersurface in four-space,” J. Geom. Symm. Phys., vol. 41, pp. 77-95, 2016.
• [29] E. Güler, H. H. Hacısalihoğlu, and Y. H. Kim, “The Gauss map and the third Laplace-Beltrami operator of the rotational hypersurface in 4-space,” Symmetry, vol. 10, no. 9, pp. 1-12, 2018.
• [30] G. Ganchev and V. Milousheva, “General rotational surfaces in the 4-dimensional Minkowski space,” Turkish J. Math., vol. 38, pp. 883-895, 2014.
• [31] A. Arvanitoyeorgos, G. Kaimakamis and M. Magid, “Lorentz hypersurfaces in E_1^4 satisfying ∆H=αH,” Illinois J. Math., vol. 53, no. 2, pp. 581-590, 2009.
• [32] K. Arslan and V. Milousheva, “Meridian surfaces of elliptic or hyperbolic type with pointwise 1-type Gauss map in Minkowski 4-space,” Taiwanese J. Math., vol. 20, no. 2, pp. 311-332, 2016.
• [33] N. C. Turgay, “Some classifications of Lorentzian surfaces with finite type Gauss map in the Minkowski 4-space,” J. Aust. Math. Soc., vol. 99, no. 3, pp. 415-427, 2015.
• [34] U. Dursun and N. C. Turgay, “Space-like surfaces in Minkowski space E_1^4 with pointwise 1-type Gauss map,” Ukr. Math. J., vol. 71, no. 1, pp. 64-80, 2019.
• [35] M. P. Do Carmo and M. Dajczer, “Rotation hypersurfaces in spaces of constant curvature,” Trans. Amer. Math. Soc., vol. 277, pp. 685-709, 1983.
• [36] L. J. Alias and N. Gürbüz, “An extension of Takashi theorem for the linearized operators of the highest order mean curvatures,” Geom. Dedicata, vol. 121, pp. 113-127, 2006.
• [37] E. Güler, Fundamental form IV and curvature formulas of the hypersphere, Malaya J. Mat., vol. 8, no. 4, pp. 2008-2011, 2020.
• [38] E. Güler and K. Yılmaz, “Hypersphere satisfying ∆x =Ax in 4-space,” Palestine J. Math., vol. 11, no. 3, pp. 21-30, 2022.